An inexact algorithm with proximal distances for variational inequalities

In this paper we introduce an inexact proximal point algorithm using proximal distances for solving variational inequality problems when the mapping is pseudomonotone or quasimonotone. Under some natural assumptions we prove that the sequence generated by the algorithm is convergent for the pseudomonotone case and assuming an extra condition on the solution set we prove the convergence for the quasimonotone case. This approach unifies the results obtained by Auslender et al. [Math Oper. Res. 24 (1999) 644-688] and Brito et al. [J. Optim. Theory Appl. 154 (2012) 217-234] and extends the convergence properties for the class of φ-divergence distances and Bregman distances.